arXiv subjects
Publications and source records attributed to Ghurumuruhan Ganesan.
In this paper we study Eulerian extensions with edge constraints and use the probabilistic method to establish sufficient conditions for a given connected graph to be a subgraph of a Eulerian graph containing $m$ edges, for a given number $m$.
In the first part of this paper, we consider weighted domination in the case where the vertices of the complete graph on~\(n\) vertices are equipped with independent and identically distributed (i.i.d.) weights. We use the probabilistic iteration to determine sufficient conditions for maximizing the weighted domination probability. In the second part, we study a weighted generalization of the chromatic number and estimate the minimum number of colours needed to satisfy the constraints when the weights themselves are random. We show that the "extra" cost incurred for weighted colouring is small if the weights have sufficiently large moments. We also consider inhomogenous random graphs where the edge probabilities are not necessarily all same and obtain bounds for the chromatic number in terms of its averaged edge probabilities.
In this paper, we study "robust" dominating sets of random graphs that retain the domination property even if a small \emph{deterministic} set of edges are removed. We motivate our study by illustrating with examples from wireless networks in harsh environments. We then use the probabilistic method and martingale difference techniques to determine sufficient conditions for the asymptotic optimality of the robust domination number. We also discuss robust domination in sparse random graphs where the number of edges grows at most linearly in the number of vertices.
In this paper, we study two problems related to planar matchings in random bipartite graphs. First, we colour each edge of the complete bipartite graph $K_{n,n}$ uniformly randomly from amongst ${r}$ colours and show that if ${r}$ grows linearly with ${n}$ then the maximum rainbow matching is a non-trivial fraction of ${r}$ with high probability, i.e. with probability converging to one as ${n \rightarrow \infty}$ Next we consider planar matchings in a dependent setting where each vertex is forced to choose exactly one neighbour from amongst all possible choices. We obtain estimates for the largest size of a planar matching and also discuss the implication of our results to longest increasing subsequences in enlarged random permutations.
In this paper, we study two examples of minimum weight random graphs with edge constraints. First we consider the complete graph on ${n}$ vertices equipped with uniformly heavy edge weights and use iteration methods to obtain deviation estimates for the minimum weight of subtrees with a given number of edges. Next we analyze edge constrained minimum weight paths in the integer lattice ${\mathbb{Z}^d}$ and employ martingale difference techniques to describe the behaviour of the scaled minimum weight in terms of the edge constraint.
In this paper, we study the redundancy of linear codes with graph constraints. First we consider linear parity check codes based on bipartite graphs with diversity and with generalized graph constraints. We describe sufficient conditions on the constraint probabilities and use the probabilistic method to obtain linear codes that achieve the Gilbert-Varshamov redundancy bound in addition to satisfying the constraints and the diversity index. In the second part we consider a generalization of graph capacity which we call as the fractional graph capacity and use the probabilistic method to determine bounds on the fractional capacity for arbitrary graphs. Specifically, we establish an upper bound in terms of the full graph capacity and a lower bound in terms of the average and maximum vertex degree of the graph.
In this paper, we obtain a Schwartz-Zippel type estimate for homogenous finite field polynomials. Specifically, we use a probabilistic recursion technique to find upper and lower bounds for the number of zeros of a homogenous polynomial and illustrate our result with two examples involving perfect matching in bipartite graphs and common zeros in a collection of polynomials, respectively.
Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In this paper we consider constrained~\(n \times n\) random matrices where each row has a deterministic number~\(r = r(n)\) zero entries and the rest of the entries are independent random variables that are positive almost surely. The positions of the zeros within each row are also random and we establish sufficient conditions for the existence of a weak law of large numbers for the permanent, appropriately scaled and centred. As a special case, we see that if~\(r\) grows faster than~\(\sqrt{n},\) then the permanent of a randomly chosen constrained~\(0-1\) matrix is concentrated around the van der Waerden bound with high probability, i.e. with probability converging to one as~\(n \rightarrow \infty.\)
In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by definition do not belong to~\(G\) and we are interested in estimating the number of bridges and the maximum bridge length, needed for constructing a Hamiltonian cycle. In our main result, we show that with high probability, i.e. with probability converging to one as~\(n \rightarrow \infty,\) we can obtain a Hamiltonian cycle with maximum bridge length a constant multiple of~\(r_n\) and containing an arbitrarily small fraction of edges as bridges. We use a combination of backbone construction and iterative cycle merging to obtain the desired Hamiltonian cycle.
For any graph~\(G,\) a set of vertices~\({\cal V}\) is said to be dominating if every vertex of~\(G\) contains at least one node of~\(G\) and separating if each vertex~\(v\) contains a unique neighbour~\(u_v \in {\cal V}\) that is adjacent to no other vertex of~\(G.\) If~\({\cal V}\) is both dominating and separating, then~\({\cal V}\) is defined to be an identification code. In this paper, we study strong identification codes with an index~\(r,\) by imposing the constraint that each vertex of~\(G\) contains at least~\(r\) unique neighbours in~\({\cal V}.\) We use the probabilistic method to study both the minimum size of strong identification codes and the existence of graphs that allow an identification code with a given index.
In this paper, we study two applications of graph minor reduction. In the first part of the paper, we introduce a variant of the boxicity, called strong boxicity, where the rectangular representation satisfies an additional condition that each rectangle contains at least one point not present in any other rectangle. We show how the strong boxicity of a graph~\(G\) can be estimated in terms of the strong boxicity of a minor~\(H\) and the number of edit operations needed to obtain~\(H\) from~\(G.\) In the second part of the paper, we consider false data injection (attack) in a flow graph~\(G\) and quantify the subsequent effect on the state of edges of~\(G\) via the \emph{edge variation factor}~\(θ.\) We use minor reduction techniques to obtain bounds on~\(θ\) in terms of the connectivity parameters of~\(G,\) when the attacker has complete knowledge of~\(G\) and also discuss stealthy attacks with partial knowledge of the flow graph.
We equip the edges of a deterministic graph $H$ with independent but not necessarily identically distributed weights and study a generalized version of matchings (i.e. a set of vertex disjoint edges) in $H$ satisfying the property that end-vertices of any two distinct edges are at least a minimum distance apart. We call such matchings as strong matchings and determine bounds on the expectation and variance of the minimum weight of a maximum strong matching. Next, we consider an inhomogenous random graph whose edge probabilities are not necessarily the same and determine bounds on the maximum size of a strong matching in terms of the averaged edge probability. We use local vertex neighbourhoods, the martingale difference method and iterative exploration techniques to obtain our desired estimates.
In this paper, we study the weight spectrum of linear codes with \emph{super-linear} field size and use the probabilistic method to show that for nearly all such codes, the corresponding weight spectrum is very close to that of a maximum distance separable (MDS) code.
Let $\mathbb{F}$ be a finite field consisting of $q$ elements and let $n \geq 1$ be an integer. In this paper, we study the size of local Kakeya sets with respect to subsets of $\mathbb{F}^{n}$ and obtain upper and lower bounds for the minimum size of a (local) Kakeya set with respect to an arbitrary set ${\mathcal T} \subseteq \mathbb{F}^{n}.$
In this paper we consider a connected planar graph $G$ and impose conditions that results in $G$ having a percolation lattice-like cellular structure. Assigning each cell of $G$ to be either occupied or vacant, we describe the outermost boundaries of star and plus connected components in $G$. We then consider the dual graph of $G$ and impose conditions under which the dual is also a percolation lattice. Finally, using $G$ and its dual, we construct vacant cell cycles surrounding occupied components and study left right crossings and bond percolation in rectangles.
In this paper we study codes for correcting deletable errors in binary words, where each bit is either retained, substituted, erased or deleted and the total number of errors is much smaller compared to the length of the codeword. We construct codes capable of correcting errors in the received codeword in real-time with small delay and low probability of decoding error.
Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less than~\(r_n,\) the adjacency distance and the resulting random graph~\(G_n\) is called a random geometric graph~(RGG). We now assign a location dependent weight to each edge of~\(G_n\) and define~\(MST_n\) to be the sum of the weights of the minimum spanning trees of all components of~\(G_n.\) For values of~\(r_n\) above the connectivity regime, we obtain upper and lower bound deviation estimates for~\(MST_n\) and~\(L^2-\)convergence of~\(MST_n\) appropriately scaled and centred.
Price discrimination for maximizing expected profit is a well-studied concept in economics and there are various methods that achieve the maximum given the user type distribution and the budget constraints. In many applications, particularly with regards to engineering and computing, it is often the case than the user type distribution is unknown or not accurately known. In this paper, we therefore propose and study a mathematical framework for price discrimination with \emph{target} profits under the contract-theoretic model. We first consider service providers with a given user type profile and determine sufficient conditions for achieving a target profit. Our proof is constructive in that it also provides a method to compute the quality-price tag menu. Next we consider a dual scenario where the offered service qualities are predetermined and describe an iterative method to obtain nominal demand values that best match the qualities offered by the service provider while achieving a target profit-user satisfaction margin. We also illustrate our methods with design examples in both cases.